What exactly is dipole movement and what is its relationship to hydrogen
bonding?
Dipole moment is due to the degree of charge separation in a molecule. How
it is calculated (or how the degree of charge separation can be determined
from a dipole moment, since that's what's obervable) is given in most
physics textbooks, but normally it's a product of the quantity of charge
and the distance between positive and negative.

All polar molecules have per
manent dipole moments, meaning that they have a positive end (usually
carbon or hydrogen) and a negative end (usually oxygen, nitrogen or a
halogen). "Temporary" dipole moments are
discussed here.

But some polar molecules are also good at coordinating highly-charged
species, such as ions. For example, ethers are good at coordinating alkali
metal cations. This is because negative charge is fairly concentrated on
the oxygen atom of the ether. But ethers are not good at coordinating
halide anions! This is because the positive charge in ethers is widely
spread among several different carbon and hydrogen atoms and doesn't
present a "concentrated target" for an anion.

charge
separation in an ether molecule

ether
molecules coordinating a lithium cation

So-called protic solvents all contain hydrogen atoms bonded directly to
electronegative atoms (oxygen or nitrogen). Since the hydrogen atom is
electropositive, it takes on a positive charge; because the hydrogen atom
is very small, the positive charge is quite concentrated. This means that
protic solvents are good at coordinating both cations and anions, because
they have both a concentrated negative charge (the oxygen or nitrogen atom)
and a concentrated positive charge (the hydrogen bonded to oxygen or
nitrogen).

charge
separation in a water molecule

water
molecules coordinating a chloride anion

The center of negative charge on one molecule can also coordinate the
center of positive charge on a neighboring molecule, if it is concentrated
enough. Because a concentrated yet non-ionic center of positive charge is
always a hydrogen atom, this is called a hydrogen bond.
Hydrogen bonds give ice its open structure and hold together the double-helix of DNA.